If we are allowed to solve the problem of accelerating bodies in Special Relativity - which postulates relativity of motion - than we can always revert the situation, and assume it is $O_1$ and $z=0$ accelerating relative to $O_2$.
In such case, and as there is always constant $v'$ (I used primed variable to denote motion) between $O_1$ and $z=0$, then regardless of how big the acceleration $a$ is, the $O_1$ will cross $z=0$ after time $t'=1s$. This will take place at a distance $x'$ between $O_1$ and $O_2$ which we can calculate based on acceleration $a$. Ultimately, we will get the dilated, albeit finite (since $t'$ is finite), $t$ for this moment from the perspective of $O_2$ .
We can then go back to our stationary $O_2$ by calculating $x$ (based on $x'$) between $O_1$ and $O_2$. This will allow us to calculate the time $t_1$ necessary for the light to travel this distance $x$. Obviously, it will be finite, and so the total time of $T=t + t_1$ will also be finite. Therefore $O_2$ will see $O_1$ reach $z=0$ in finite time $T$.
If someone says we cannot do that, than we simply cannot apply SR to this situation, which means we do not have SR paradox here.
EDIT: In the first sentence of my answer I wrote: "If we are allowed to solve the problem of accelerating bodies in SR ...". There are numerous claims (not only on this forum) that acceleration can be easily and rightfully handled by SR, so I decided to show what happens, when you do that.
Now, I personally prefer to stick to Einstein's postulate which means: "Thou shall not consider non-inertial frames in Special Relativity":
"If, relative to K, K' is a uniformly moving co-ordinate system devoid of rotation, then natural phenomena run their course with respect to K' according to exactly the same general laws as with respect to K." And then: "In order to attain the greatest possible clearness, let us return to our example of the railway carriage supposed to be travelling uniformly. We call its motion a uniform translation ('uniform' because it is of constant velocity and direction."
However, in the discussion below John Rennie maintains that acceleration can be rightfully considered in SR, and also that by doing so we invalidate the postulate that no frame of reference is preferred. So we get rid of two basic postulates of SR (inertial frames and no preferred frames), and we still keep calling it SR? To me it's like putting up a cow with a plate "Cow", and then replacing the cow with a goat, but still keeping the same plate. Excuse my trivial example, but that's how I see it.
John Rennie even cited John Baez in his own answer to the question (now removed). However, if one follows this link and clicks on the "accelerating clocks", they will find this as explanation: "... the accelerated clock's rate is identical to the clock rate in a 'momentarily comoving inertial frame' (MCIF), which we can imagine is holding an inertial clock that for a brief moment slows to a stop alongside the accelerated clock, so that their relative velocity is momentarily zero. At that moment they are ticking at the same rate. A moment later, the accelerated clock has a new MCIF, again one that is moving momentarily to match its speed, and there is a new inertial clock that briefly slows to a stop alongside the accelerated clock." Which means, in plain English, that the clock is stopped, and yet at the same time it is ticking. Now, that's not SR, that's SF to me ... (I have seen two other explanations for acceleration in SR on this forum, and they both used either exactly the same or a very similar trick).
Einstein, when deriving his field equations for GR said (page 98) here: "For infinitely small four-dimensional regions the theory of relativity in the restricted sense is appropriate, if the coordinates are suitably chosen." "Relativity in the restricted sense" is simply Special Relativity. So he believed he needed to go down to "infinitely small regions" in order to get rid of acceleration (i.e. gravity - which he - through his equivalence principle - postulates is the same to prove his GR theory), and be allowed to use SR. And he also said in this same book (page 90): "By the word special it is signified that the principle [of relativity] is limited to the case, where K' has uniform translatory motion with reference to K". Here we go! Special, because there are no accelerations. If we introduce accelerations, we are no longer on the grounds of "relativity in the restricted sense" called "special".
This is not to say that what Einstein had said 100 years ago cannot ever be questioned. He is no god whatsoever, and science moves on. I have my own doubts about various of his claims. But then, if one wants to use his theory, and yet get rid of his basic postulates, then he needs to show it is a valid move. And, obviously, I'm not saying accelerations cannot be considered by physics. Sure they can. But in order to claim it can be done on the grounds of SR, one must simply prove it. I must see it to believe it.
What I actually proved in my answer is that if we introduce acceleration to SR, and yet do what the theory allows us to do - i.e switch the frames of reference - then we will obtain two different results. The interpretation of this fact seems just all to obvious.